Select the option that is related to the third number in the same way as the second number is related to the first number.

223 : 350 :: 519 : ?

To find the missing number in the analogy, let us analyze the relationship between the first pair of numbers: 223 and 350.

First, we examine the connection between 223 and perfect cubes:
The cube of 6 is:
$6^3=216$
If we add 7 to 216, we get:
$216+7=223$

Next, let us look at the second number, 350:
The cube of 7 is:
$7^3=343$
If we add 7 to 343, we get:
$343+7=350$

Thus, the relationship pattern between the first two numbers is:
$(n^3+7):({(n+1)}^3+7)$
where $n=6$.

Now, let us apply this pattern to the third number, 519:
We find the closest perfect cube below 519, which is the cube of 8:
$8^3=512$
Adding 7 to 512 gives:
$512+7=519$
Here, $n=8$.

Following the established pattern, the next term must correspond to $n+1=8+1=9$:
$9^3+7$
Calculating the cube of 9:
$9^3=729$
Adding 7 to 729 gives:
$729+7=736$

Therefore, the missing number is 736.